Lesson 5

Graphs, Tables, and Equations

These materials, when encountered before Algebra 1, Unit 2, Lesson 5 support success in that lesson.

5.1: Math Talk: Solving Equations (10 minutes)

Warm-up

The purpose of this Math Talk is to elicit strategies students have for solving equations in one variable, especially equations involving an expression in parentheses being multiplied by a coefficient. Developing fluency with these types of equations will be helpful in the associated Algebra 1 lesson when students substitute expressions in place of single variables and solve equations with expressions in parentheses.

In particular, students should see that to solve an equation like $$100=10(x-5)$$, a productive first step is to divide each side by 10.

Math Talks build fluency by encouraging students to think about expressions or equations and rely on what they know about properties of operations and equality to mentally solve a problem.

Launch

Display one problem at a time. Give students quiet think time for each problem and ask them to give a signal when they have an answer and a strategy. Keep all problems displayed throughout the talk. Follow with a whole-class discussion.

Student Facing

Solve each equation mentally.

$$100=10(x-5)$$

$$300=30(x-5)$$

$$15-971=x-4-971$$

$$\frac{10}{7}=\frac17 (x-19)$$

Activity Synthesis

Ask students to share their strategies for each problem. Record and display their responses for all to see. To involve more students in the conversation, consider asking:

• “Who can restate $$\underline{\hspace{.5in}}$$’s reasoning in a different way?”
• “Did anyone have the same strategy but would explain it differently?”
• “Did anyone solve the problem in a different way?”
• “Does anyone want to add on to $$\underline{\hspace{.5in}}$$’s strategy?”
• “Do you agree or disagree? Why?”

5.2: On the Line (15 minutes)

Activity

The purpose of this activity is to recall the connection between equations and graphs. This will be useful when students create an equation to represent a situation and graph it in an associated Algebra 1 lesson.

Launch

Allow students to use technology to graph the equations in the task statement.

Student Facing

1. Sketch a graph representing each of these equations.
1. $$y=2x$$
2. $$y = \frac{1}{2}x$$
3. $$y=x+2$$
2. For each point, which graph or graphs is it on? How can you tell by using the equation?
1. $$(1,3)$$
2. ​​​​​​​$$(0,0)$$
3. ​​​​​​​$$(3,6)$$
4. ​​​​​​​$$(3,1.5)$$

Activity Synthesis

The purpose of this discussion is to highlight the connection between an equation and the graph. Here are sample questions to promote class discussion:

• "How can you determine which graphs include a specific ordered pair?" (I search for the ordered pair on the coordinate plane, and if it appears on the line, then I know that it is a solution for the respective equation.)
• "How do the coordinates for ordered pairs on a given line relate to the line's equation?" (The coordinates are solutions to the equations of the line.)
• "Could you determine which graphs include specific ordered pairs without using the graph?" (Yes, you can use the equation to determine if an ordered pair lies on the respective line. If substituting the coordinates into an equation makes the equation false, then we know that particular ordered pair does not lie on the line.)

5.3: Take Turns: Graphs, Tables, Equations, and Situations (15 minutes)

Activity

In this activity, students get a chance to practice applying their skills at connecting multiple representations of the same linear relationship.

They can use the structure of the card-matching task to check their thinking and make sense of the concepts they are practicing, as wrong matches will make some piles uneven or a representation that seems to match one representation in a pile may not match another.

The practice will pay off in the associated Algebra 1 lesson when they create graphs and equations to describe the relationship between two quantities from a situation and match them with an equation.

Launch

Arrange students in groups of 2 and distribute a set of cut-up slips to each group. (Note that you should reuse the same slips from lesson 3, except include the graphs this time.) Ask students to re-do the matches from lesson 3; you can tell them the correct matches to expedite the process. Ask students to take turns: the first partner identifies a graph to match the already matched equation, table, and situation and explains why they think the graph belongs, while the other listens and works to understand. When both partners agree on the match, they switch roles.

Student Facing

1. Take turns with your partner to match a graph with each set of matching cards. Eventually all the cards will be sorted into groups of 4 cards (an equation, situation, table, and graph).

2. For each match that you find, explain to your partner how you know it’s a match. Ask your partner if they agree with your thinking.

3. For each match that your partner finds, listen carefully to their explanation. If you disagree, discuss your thinking and work to reach an agreement.

Activity Synthesis

The goal of this discussion is to help students understand that rows in tables and points on lines represent solutions to the related linear equation.

Display the graph, table, and equation for Mai’s running speeds.

minutes miles
10 0.4
25 1
60 2.4
90 3.6

$$\displaystyle y = 0.04x$$

Possible questions for discussion:

• If $$x$$, is 10, what does $$y$$ equal? How do you know? (0.4 because $$10 \boldcdot 0.04= 0.4$$.)
• What does the point $$(10,0.4)$$ mean in the story? (Mai can run 0.4 miles in 10 minutes.)
• How can you see from the graph what $$y$$ is if $$x$$ is 10? (I can find the ordered pair on the graph and see that when $$x$$ is 10, $$y$$ is 0.4.)
• How can you see from the table what $$y$$ is if $$x$$ is 10? (I can see that 10 minutes is paired with 0.4 miles.)
• How can you use the equation to find out what $$y$$ is if $$x$$ is 10? (I can substitute 10 for $$x$$ in the equation $$y=0.04x$$ and solve for $$y$$.)
• When would you choose to use a graph to tell you an $$x$$ value if you know the $$y$$ value? When would you use a table? A graph?
• If $$y$$ is 3.6, what does $$x$$ equal? How do you know? (90 because $$90 \boldcdot0.04= 3.6$$)
• What does the point $$(90, 3.6)$$ mean in the story? (Jada can run 3.6 miles in 90 minutes.)
• How can you see from the graph what $$x$$ is if $$y$$ is 3.6? (I can find the ordered pair on the graph and see that when $$y$$ is 3.6, $$x$$ is 90.)
• How can you see from the table what $$x$$ is if $$y$$ is 3.6? (I can see that 3.6 miles is paired with 90 minutes.)
• How can you use the equation to find out what $$x$$ is if $$y$$ is 3.6? (I can substitute 3.6 for $$y$$ in the equation $$y=0.04x$$ and solve for $$x$$.)